Cooling flow behavior is investigated within the multiple serpentine passages with turbulators on the leading and trailing walls of an axial gas turbine blade operating at design-corrected conditions with accurate external flow conditions. Pressure and temperature measurements at midspan within the passages are obtained using miniature butt-welded thermocouples and miniature Kulite pressure transducers. These measurements, as well as airfoil surface pressure field data from a full computational fluid dynamics (CFD) simulation, are used as boundary conditions for a model that provides quantitative values of film-cooling blowing ratio for each film-cooling hole on the blade. The model accounts for the continuously changing cross-sectional area and shape of the channels, frictional pressure loss, convective heat transfer from the solid portion of the blade, massflow reduction as coolant bleeds out through film-cooling or impingement holes, compressibility effects, and the effects of blade rotation. The results of the model provide detailed coolant ejection information for a film-cooled rotating turbine airfoil operating at design-corrected conditions and also account for the highly variable freestream conditions on the airfoil. While these values are commonly known for simpler experimental geometries, they have previously either been unknown or estimated crudely for full-stage experiments of this nature. The better-quantified cooling parameters provide a bridge for better comparison with the wealth of film-cooling work already reported for simplified geometries. The calculation also shows the significant range in blowing ratio that can arise even among a single row of cooling holes associated with one of the turbulated passages, due to significant changes in both coolant and local freestream massfluxes.

## Introduction

The majority of these experiments are performed using simplified, nonrotating geometries, such as flat plates or linear cascades. Film-cooling parameters such as blowing ratio can be calculated from quantities that are relatively easy to measure for these simplified geometries. Unfortunately, rotating turbine experiments present much more difficulty in quantifying even basic cooling parameters such as total coolant massflow into the blade due to the complexity of the flow path. Often, as is the case for this turbine, the coolant provided for purge cooling and blade cooling is supplied to the rig together, and it is impossible to directly measure how much flow goes to the blades, and how much goes to the purge without having flow meters on the rotating blade. Additionally, the interface between rotating and stationary sections requires at least some leakage of coolant through bearings and seals that is very difficult to quantify. And even if a good measurement or estimation of coolant to a serpentine passage is obtained, the amount of coolant that gets ejected through each hole is not the same. The pressure, density, and velocity of the coolant within the serpentine passages are highly variable, as are the conditions on the local airfoil surface at the outlets of the cooling holes. The airfoil surface static pressure can vary by a factor of 2, and the freestream mass flux (as measured just outside the boundary layer) can vary by as much as a factor of 3.

One of the first film-cooled rotating experiments was performed by Dring et al. [2]. Blowing ratios were provided for the experiment but there was only a single hole on both the pressure and suction-sides. Abhari and Epstein [3] performed an experiment with five rows of cooling holes and quantified average blowing and momentum ratios for two of the rows but did not quantify blowing for the other rows or investigate spanwise discrepancies in cooling flows within a row. A more recent experiment reported by Haldeman et al. [4] quantifies total rotor cooling as a percentage of total core flow but does not quantify cooling flows on a row-by-row basis or provide any blowing ratio values.

The first half of this paper presents a model that is used to estimate the coolant properties within each serpentine passage of a high-pressure turbine blade as a function of span. These properties, combined with external airfoil pressures, are then used to estimate the ejection massflow for each hole. This model is unique in that it uses temperature and pressure measurements taken at midspan of several of the serpentine passages for a full-stage experiment as boundary conditions. This provides a much more accurate and local boundary condition than prior works, which rely on accurate modeling of flow through several sequential passages and turns. In addition to the internal data, a CFD prediction of the external pressure field at each cooling hole is used. Many of the details of this model are specific to this particular serpentine configuration but the overall methodology should be applicable to other configurations as well.

The second half of this paper presents the results of this model for the aforementioned experiment of a high-pressure transonic turbine. These results show the overall variation of cooling rates even within the same row of holes being fed by the same internal passage, as well as the variation between rows, and with variable amounts of total coolant supplied. The model goes on to show that, while coolant ejection massflux varies by more than a factor of 2 across the different holes, the freestream massflux varies by a factor of 4 and is actually the more dominant factor in determining local blowing ratios.

## Experimental Setup

The experiment that provided the boundary conditions for this model generated an extensive dataset, the vast majority of which is beyond the scope of this paper. A more complete description of the experiment is provided in the accompanying papers by Nickol et al. [5]. This paper will focus on calculating the properties of coolant within the serpentine passages using the measurements obtained from the experiments.

The rotor blade used is engine hardware from a modern Honeywell high-pressure turbine stage. Photographs of the blade suction and pressure sides are provided in Fig. 1, and a schematic of the airfoil and internal passages is provided in Fig. 2. There are six coolant passages numbered from 1 at the leading edge to 6 near the trailing edge. Passages 2, 4, 5, and 6 (underlined in the schematic) are instrumented to obtain a static temperature and static pressure measurement at midspan. There are seven rows of cooling holes labeled A–G, starting with the aft row of holes on the pressure side and going clockwise around the blade (as shown in Figs. 1 and 2). Only passages 2 and 3 are fed coolant at the root of the blade, and they are supplied by separate channels. Passage 2 continuously feeds into passage 1 via 19 impingement holes, and passage 1 feeds 76 showerhead cooling holes in five rows (B–F). Passage 3 does not have any cooling holes but there are holes in the blade tip in the turn segment between passages 3 and 4. The coolant from passage 3 that does not get ejected at the tip continues from the tip back down to the hub through passage 4, where row G, containing nine cooling holes, ejects to the blade suction-side. The remaining coolant gets turned again at the blade hub and travels up passage 5, where row A ejects through 16 cooling holes to the blade pressure side. The remaining coolant once again turns at the tip and heads down passage 6, which feeds a pin-fin cooling area near and trailing edge slots.

The six passages in the blade are significantly different from each other in geometry. Cross-sectional area, hydraulic diameter, and even shape all vary from row-to-row (Fig. 2 is not to scale but is roughly accurate in terms of shape and relative size) and as a function of span. Table 1 provides a summary of some of the properties for each cooling passage including the cross-sectional area and hydraulic diameter relative to passage 2, the turbulator height and spacing ratios, and the rotation number. Passage 2 sees the most significant change in properties over the length of the passage, so both hub and tip values are provided. For areas and hydraulic diameters, the value is normalized by the passage 2 hub value.

Passages 2–6 all feature square-cross-sectioned turbulators on the leading and trailing walls that are oriented perpendicular to the direction of flow. The turbulator pitch-to-height ratio (*P*/*e*) is 11.75. The turbulator blockage ratio (*e*/*D _{h}*) is not constant as the hydraulic diameter varies both with span, and from passage to passage, and average values are given in Table 1.

A rotation number is not calculated for passage 1 because it is fed through 19 impingement holes across the entire span rather than at one end. The flow spreads between the jets at a low bulk velocity and in nonuniform directions, causing the rotation number to approach infinity. Additionally, passage 1 does not contain turbulators.

The turbine is setup such that one circuit of cooling flow is injected through the tangential on-board injector (TOBI) to supply the blade film-cooling flow, forward purge cooling, and some leakage flow through the bearings and seals. This combined TOBI cooling massflow rate is measured using two Emerson Micro Motion Coriolis massflow meters operating in parallel. The TOBI flow rate is then normalized by the total turbine-inlet massflow for this experiment and varies from 5.28% to 8.64% across the run matrix.

The experimental uncertainty in the internal pressure measurements is approximately ±1.4 kPa or 0.47% of inlet pressure. The internal temperatures have uncertainties within ±0.5 K.

## Model Theory

The basic format of the model is similar to CFD in that it breaks each serpentine passage into a series of nodes and defines the complete thermodynamic state, velocity, and massflow of coolant at each node. There is assumed to be no variation in these values within each node (with the exception of a small pitchwise pressure correction that is explained later).

A stencil of the model is provided in Fig. 3. Passages 1, 4, and 5 feature cooling holes at many of the nodes, where a certain amount of massflow is bled off. Because passage 1 supplies the five-row showerhead, some nodes feed multiple cooling holes. Passage 2 contains one large cooling hole at the tip but otherwise supplies passage 1 through 19 separate impingement passages. One unique aspect of this model is that there is no defined massflow, pressure, or temperature condition at any inlet or outlet. The boundary conditions are defined exclusively at midspan where data exist.

Total temperature in the cooling passages is not a constant due to heat transfer from the metal into the blade. The heat-flux into the coolant can be expressed in two ways shown in Eq. (8), with the first being the definition of the (unknown) convective heat transfer coefficient *h*, and the second being the result of a simple energy balance. Combining these two forms gives Eq. (9). This equation applies to every node except *i* = 1, with a slightly modified equation is used to link the hub (*i* = 1) temperature of passage 5 to that of passage 4.

These equations are combined with the two temperature measurements at midspan and result in one extra equation. This extra equation is combined with Eq. (10) to give a single value of Nu* _{i}*/Nu

_{∞}for the entire blade, which is defined by taking the local Nusselt number and normalizing by the Nusselt number for fully developed turbulent flow according to the Dittus–Boelter/McAdams equation [6]. This constant is then fed into a similar set of equations for passages 1 and 2.

*r*is defined positive out, regardless of whether the bulk flow is moving toward or away from the hub)

Frictional losses require a value for the friction factor, which is defined in Eq. (13), where *L* is the length of the channel producing the pressure-drop Δ*p _{f}*. The instrumentation for this experiment does not allow for direct measurement or calculation of the friction factor but many past studies focusing on internal passage flow have published values of channel friction factor. These are often normalized by the friction factor for a turbulent circular channel ($f\u221e$), which is calculated using a fit of the Kármán–Nikuradse equation, as shown in Eq. (14). The work of Chandra et al. [7] on similar but stationary walls provides approximate values for $f/f\u221e$ for each channel. Liou et al. [8] explore correlations for friction factor for smooth channels as a function of Reynolds and rotation numbers and show that the channel-average friction factor is increased by 10–20% due to rotation numbers on the order of those seen in this experiment. Combining the results of these two studies produces final values of $f/f\u221e$ of 1.1, 4.5, 6.6, and 5.0 for passages 1, 2, 4, and 5, respectively. It is difficult to determine how accurate these values of friction factor are but the final solution is not very sensitive to them; halving or doubling these values results in a ±1.2% change in blowing ratio at most.

Equations (13) and (14) can be combined to produce an expression for the pressure gradient due to friction as a function of $f/f\u221e$, as shown in Eq. (15). The sign of the *w* parameter is included because the bulk flow may be traveling radially inward or outward, and this will determine the direction of pressure loss.

*i*= 1. For the forward circuit, the system is closed by the one pressure measurement in passage 2, and an extra equation from the mass balance that will be explained in a later paragraph. The aft circuit is closed by pressure measurements in both passages 4 and 5, and then over-constrained by a hub-turn pressure-drop equation that couples the hub nodes of passages 4 and 5. This extra equation is actually required by the mass balance, also explained in a later paragraph

Equation (22) also requires a discharge coefficient for each cooling hole. This coefficient would be nearly impossible to measure directly on a rotating machine but fortunately there are dozens of previously published experiments designed to measure discharge coefficient for a variety of flow conditions and geometries (see Ref. [9] for a review of such work). For the central three rows of the showerhead, *C _{d}* = 0.65 is used, as per Ref. [10]. The extreme pressure and suction-side showerhead rows used

*C*= 0.60 and

_{d}*C*= 0.72, respectively, as per Ref. [11]. The remaining two rows, one on the suction-side, and the other on the pressure side, both use

_{d}*C*= 0.70, again from Ref. [11].

_{d}## Details of the Accompanying Computational Fluid Dynamics

The internal flow model presented in this paper requires an accompanying 3D full-stage CFD prediction to provide cooling hole outlet boundary conditions. A full description of the computational setup, as well as an analysis of its results on the airfoil, is provided in an accompanying paper [5] but a brief description of its setup is provided here for completeness.

The CFD is a 3D steady RANS simulation. A schematic of the full computational domain is provided in Fig. 4. The domain begins approximately one stator vane chord upstream of the stator and includes one stator vane, one rotor blade, and an exit region (also about one blade-chord long). The interface between stator and rotor utilizes a mixing plane approach, and periodic boundary conditions are applied in the pitch direction. Two cooling circuits are meshed: the aft purge which is not of concern in this paper and the rotor coolant, which is set by the measured TOBI flow minus leakage and supplies both the blade flow and forward purge. Coolant and core flow inlet boundary conditions are prescribed by mass flow rates, and the exit boundary condition is set by static pressure. The computation utilizes a *k–ε* turbulence model and is both meshed and solved using the star-ccm+ suite.

## Validation of the Model

The model presented in this paper makes several assumptions and simplifications, and therefore it is important to validate its end results where possible. The simplest comparison is to compare overall massflows. Recall that the model does not utilize massflow measurements in any way—the passage flow rates are calculated entirely from the pressure measurements in passages 2, 4, and 5, the geometry of the passages, and the friction factor. Also recall that the TOBI flow measurement is taken upstream in the cooling system before it enters the rig, and this flow splits to the blade cooling flow, the forward purge flow, and the leakage flow. A proprietary Honeywell flow model is used to determine the massflow split among these three destinations based on experimental pressure measurements from the TOBI region, purge cavity, and seals. This model predicts that between 60% and 61% of the TOBI flow will go through the blade for each run.

Table 2 compares the coolant flow rate obtained from the experimental mass flow measurement for a 60.5% flow split to the flow rate predicted by the model described in this paper based on internal pressure measurements and flow friction. The results agree well with a maximum deviation of 0.28% of core flow or a relative difference of 7.3%.

## Results

A plot of the internal static pressure as a function of span for the 6.85% flow case for the four modeled passages is provided in Fig. 5, with the three measured data points shown with large + symbols. The most immediate conclusion is that the pressure increases with span, as the centrifugal pressure gradient dominates all four passages. Passages 1 and 2 are held with a fairly constant difference between them, as 2 continuously feeds into 1. Passages 4 and 5 have differing slopes because of the change in flow direction. The bulk flow in passage 4 travels from high to low span (as shown by the arrow), so friction works with the centrifugal forces to cause the pressure at high-span to be greater. Friction in passage 5 on the other hand causes a reduction in pressure at higher span. Also, the cross-sectional area of passage 5 is also about half that of passage 4, resulting in approximately double the flow velocity and four-times the frictional pressure loss.

Ultimately solving for the pressure is just an intermediate step to quantifying the coolant flow rates. Figure 6 presents the global blowing ratio for the nominal cooling rate at each cooling hole, with different symbols for each row of cooling hole (refer to Figs. 1 and 2 for the row definitions). As expected, rows E, F, and G, the rows on the suction-side, provide the greatest global blowing ratio because the external pressure on the suction-side is significantly lower than at the leading edge or on the pressure side. Row F is the greatest, above row G because the internal pressure of passage 1 is far greater than passage 4, as seen in Fig. 5, and above E because of a lower external pressure. The massflux at high-span in row F is more than double that on the pressure side (row A).

It is also worth noting some of the spanwise trends: coolant massflux increases at higher span as the coolant supply pressure increases with the exception of row A. This is actually a little misleading, as it is not at a constant value of wetted distance. Figure 1 shows that this row of holes bends aft at low span, which results in a lower external pressure and therefore higher ejection rate. This can further be seen in Fig. 7, which shows the pressure at the outlet of each cooling hole (normalized by stator inlet total pressure). The pressure at the outlet of cooling holes of row A drops quickly at low span.

Figure 8 shows the local blowing ratio in a similar manner as Fig. 6; however, the results are much different. At low span, row D presents the greatest blowing ratio, 2–3 times that of any other row. Moving up in span, row D falls, while rows B and C increase in blowing ratio. This is caused by changes in freestream massflux. At low span, the stagnation point of the airfoil is close to the cooling holes at row D, resulting in very low freestream massfluxes and therefore very high blowing ratio. At higher span, the stagnation point moves toward the pressure side, toward rows C and B. This decreases the freestream massflux for the higher-span holes in those rows, and they experience higher blowing ratio, while row D effectively becomes a suction-side row. This is perhaps better illustrated by Fig. 9, which plots local freestream massflux at each hole outlet, normalized by the global average. The first hole (lowest-span) in row D sees very low freestream massflux but the midspan holes see about double the first hole's massflux, with the highest-span hole seeing triple. Conversely, the lowest-span hole in rows B and C experiences triple the freestream massflux as the highest-span hole from those rows.

The local blowing ratios for rows A, E, F, and G are much lower and more consistent, both from row-to-row, and as a function of span within a row. In general, increases in coolant massflux are caused by lower external pressures. Lower external pressures are often the result of an accelerated flow, meaning greater freestream massflux. In essence, away from the stagnation point, the freestream and coolant massfluxes tend to both move up or down together. The spanwise trends are important to note but they remain mostly unchanged as the overall cooling flow is increased or decreased. Consequentially, the remaining figures that show the effect of variable cooling will only provide the average blowing rate for each row.

Figure 10 shows how the local blowing ratio acts as a function of cooling flow provided. As would be expected, the blowing ratio increases for all rows as additional coolant is provided but the degree to which the flow increases for each row is not uniform. At low coolant flow, row A features the lowest local blowing ratio, just below the suction-side rows E, F, and G. The other rows closer to the stagnation region display the greatest blowing rates. As the total cooling increases, row A increases in blowing ratio more rapidly than the suction rows (E–G) and actually surpasses them by the greatest coolant massflow. The leading edge rows (B–D) also increase rapidly. Ultimately, rows E–G increase in blowing ratio by about 25% from the greatest cooling flow rate to the lowest, while rows A–D approximately double in blowing ratio.

The difference in behavior can be explained through looking at the flow characteristics at each row from a basic compressible flow point-of-view. Equation (20) calculates the isentropic ejection Mach number, the maximum Mach number the coolant could reach being ejected through a nozzle based on the total supply pressure and static outlet pressure (assuming isentropic flow). The row-averages for this are plotted in Fig. 11 for each cooling flow rate. Rows F and G are at Mach number greater than unity (choked) even for the lowest cooling rate, and row E is close, becoming greater than unity (choked) by the highest cooling rates.

Because rows F and G are choked, the only way their cooling massflows can be increased is to increase the coolant supply pressure (see Eq. (22)) as total coolant flow is increased. The other rows see the same massflow increase due to supply pressure but also experience additional increased coolant ejection because the increased ejection Mach number. The effect does have diminishing returns, however. Row A sees an ejection Mach number increase from about 0.31 to 0.72 from the lowest to the greatest cooling flow rates, resulting in an ejection increase of 83% of the low-flow value just from the increased Mach number. Row E's ejection Mach number increases from about 0.85 to 1.15 but that increase only causes a 2% increase in massflow. This is illustrated graphically in Fig. 12 and is the reason that row E appears very similar to F and G in Fig. 11.

## Conclusions

A model is created to estimate cooling flow rates out of each of 101 film-cooling holes on a high-pressure turbine rotor blade. Static pressure and temperature measurements are taken at midspan within the serpentine cooling passages, and these measurements are used with a CFD prediction of the external airfoil flow field as the boundary conditions. The primary goal of this model is to accurately quantify the blowing ratio for each cooling hole for a rotating turbine experiment without running a full-stage cooled CFD prediction.

The model successfully quantifies coolant massflow rates through each of the 101 cooling holes for the full run matrix of different total coolant rates. These are then used to calculate blowing ratio according to both stage-averaged freestream massflux (global blowing ratio) and CFD-determined local freestream massflux for each cooling hole (local blowing ratio). This turns out to be a very important distinction. The massflux out of suction-side holes can exceed twice that of holes on the pressure side but their physical (local) blowing ratios are fairly close due to the greater freestream massflux over the suction-side. It turns out that the greatest physical (local) blowing ratio is not caused by high coolant massflux, but by low freestream massflux, and occurs near the stagnation point at the leading edge. Additionally, because the stagnation point moves from low to midspan, the cooling row that experiences the greatest blowing ratio changes as well. These are important results: accurate values for blowing ratio on cascade and full-stage experiments require accurate values for local freestream massflux at each cooling hole.

Increasing the overall total coolant massflow through the blade increases blowing ratio for each hole as well but the effect is not equal. Holes on the pressure side and leading edge eject a greater share of the added coolant because holes on the suction-side are already choked or near-choked even at low cooling rates. This is also an important consideration. Even a small increase in airfoil coolant flow rate can incur a significant penalty on overall engine performance but it is possible that the majority of the added coolant may not end up where it is needed. Another consideration is the effect of having transonic flow within cooling holes, especially as many designs move to diffuser-shaped cooling holes. If the entry region of the cooling hole is choked, the hole profile may first act as a supersonic nozzle, rather than subsonic diffuser, and there is a likelihood for standing normal shocks within each cooling hole.

## Acknowledgment

The authors would like to thank Honeywell Aerospace for providing the engine hardware and much of the funding for this experiment. Further funding was provided by the U.S. Army through the AGGT program and the U.S. Federal Aviation Administration (FAA) through the CLEEN program. In addition, the Gas Turbine Lab technical staff including Jeff Barton, Ken Fout, Igor Ilyin, and Jonny Lutz were essential in the design, building, and instrumenting of the rig. Finally, we would like to thank students Hannah Lawson, Tim Lawler, Chris Cosher, Matt Tomko, Miles Reagans, Eric Barbe, and Kevin McManus for support through miscellaneous tasks in the setup of the experiment.

## Nomenclature

- $A$ =
cross-sectional area

- BR =
blowing ratio

- $Dh$ =
serpentine passage hydraulic diameter

- $e$ =
turbulator height

- $h$ =
convective heat transfer coefficient

- $m\u02d9$ =
massflow rate

- Ma =
(relative) Mach number

- $p$ =
pressure

- $P$ =
turbulator pitch

- $r$ =
radial (spanwise) distance from axis-of-rotation

- Ro =
serpentine passage rotation number

- $T$ =
temperature

- TOBI =
tangential on-board injector (combined source of blade and purge coolants, as well as leakage coolant)

- $w$ =
(relative) bulk flow velocity

- $y$ =
tangential (pitchwise) coordinate

- $\Omega $ =
rotational velocity (rad/s)